Interference is the property by which waves of the same frequency, but travelling in different directions, superimpose in such a manner as to give alternate regions of stationary high and low, or zero, intensities. In the case of light, if two or more light waves from different sources are superimposed, no simple description of the observed phenomenon is possible in general since the wave fluctuations are usually uncorrelated. However, if two waves from a single monochromatic point source are superimposed, the intensity of the light varies in region of superposition and, under the proper conditions, interference may be observed in the form of an interference pattern of fringes of alternating intensity. Interferometric fringe patterns are useful in engineering and science. For example, interferometers are used to quantify surface microtopography (interference microscopy) and for the measurement of refractive indices, displacements, strains, optical characteristics and fluid properties. In industrial applications, fringe patterns are useful in inspection and quality control. In such industrial applications, it would be desirable to rapidly and accurately reconstruct three-dimensional surfaces and objects from two-dimensional fringe patterns.
Several methods exist for producing fringe patterns by optical techniques. Interference patterns are produced by the use of interferometers. All interferometers divide a beam of light into two or more parts which are directed along different optical paths reflected and then recombined to form an interference pattern. In effect, interferometers measure differences in the optical path, e.g., a difference in the geometric path can be measured when the two beams travel in the same medium or a difference in the refractive index can be measured when the geometric paths are equal, but the beams travel in different media.
In the case of a two-beam interferometer, if the objects, e.g., planar mirrors, from which the two beams are reflected are perpendicular to the beams, the path lengths are equal and the media through which the beams travel are the same, the recombined light waves will constructively reinforce each other and a single bright image will be formed. If the path lengths differ by one-quarter of the wavelength (.lambda.) of the light used, destructive interference will occur and the viewing field will appear dark. If one mirror is slightly tilted with respect to the other, a series of parallel fringes representing height differences of .lambda./2 will be observed. If one of the mirrors is replaced by the reflecting surface of an object to be examined, height differences present on the object will cause perturbations in the fringe pattern. By properly interpreting the interference pattern, quantitative differences in height can be determined.
Interferometers are precision measuring instruments. For example, the vertical resolution of two-beam interferometry is limited by the sinusoidal intensity distribution of the fringes. Fringe displacements of about one-fifth of a fringe spacing, which corresponds to 500 .ANG. of vertical height, can be readily estimated visually. The use of high-contrast photography may further improve the vertical resolution of two-beam interferometry to the 100 .ANG. range. In multiple-beam interferometry, the light reflects back and forth several times between an optical flat and the object surface. Under ideal conditions, surface displacements as small as 1/1000 of the wavelength, or 5 .ANG., may be detectable on highly reflective surfaces.
Interferometers are not the only devices by which useful fringe patterns may be obtained using optical techniques. Another useful but less sensitive, technique than interferometry is moire contouring. Moire contouring is particularly useful where the objects are large and the surfaces to be measured are too rough, or both, to generate suitable contour fringes by interferometry. In one moire technique, the object is illuminated through a Ronchi grating, consisting of alternate opaque and transparent lines of equal width d, by a collimated light source at an angle .theta. to the grating to produce an array of shadows on the object. If the object is also viewed through the grating at an angle .theta., the array of shadows interferes with the Ronchi grating to yield a moire pattern consisting of alternating fringes. The spacing between adjacent fringes represents a height difference h given by h=d/(tan .theta.).
Another moire technique is the scanning moire method described by Idesawa et al. in Applied Optics, vol. 16, no. 8 (August 1977) at pages 2152-2162, the entirety of which is herein incorporated by reference, wherein the grating image is projected onto the object and the fringes are formed by electronic scanning and sampling techniques in a TV camera, thus replacing the conventional reference grating with a virtual grating.
It is often difficult to extract information from an interference or fringe pattern because there is insufficient information in the unlabelled fringe image, e.g., there is insufficient information on a single pattern to determine whether a surface feature is an elevation or depression. An interpretation may be obtained with an interferometer by slightly displacing the reference mirror of the interferometer, by changing the focus, or moving the object and then photographing another pattern. The direction of motion of the fringes may be found by comparing the two patterns. For example, as the surface is displaced toward the reference mirror, fringes of an elevation will be seen to originate at the point of maximum height and move laterally outward from that point. Alternatively, fringes contouring a depression will collapse into the region of maximum depth as the surface is displaced toward the reference mirror.
Although other techniques, such as the use of stereo pairs and double exposures, exist to aid in the interpretation of fringe patterns, the quantitative analysis of fringe patterns can be very tedious and in some cases can yield ambiguous results. Thus, there is considerable interest which has been increasing recently in automating the process of fringe analysis including the reconstruction of three-dimensional surfaces and objects from two-dimensional fringe patterns. In the case of the scanning moire method of Idesawa et al. discussed above, the pitch and rotation of the virtual grating can be varied electronically to determine the sign of the contour lines, thus rendering unnecessary the use of other techniques such as moving the object to interpret the sense of the contour lines.
Automation, however, has not as yet been able to solve some of the more complex problems associated with interference fringe and moire pattern analysis. For example, for topographical structures known as saddle points, where the principal curvatures are of the opposite sense, the contour fringes may touch and intersect. Branched fringes that touch and intersect pose particularly difficult problems to current automated systems as do fringes that end at surface discontinuities. Automation also introduces its own sources of difficulties, e.g., signal noise and lack of contrast in images processed through TV cameras.
Generally, current automated interpretation technology accommodates the above-mentioned difficulties with intersecting and touching fringes by pausing or obscuring the area of difficulty and calling for operator intervention. Funnell, in the September 1981 issue of Applied Optics, vol. 20, no. 18, the entirety of which is incorporated herein by reference, describes one such interactive system. Once some initial parameters are set, the described system is capable of proceeding with the analysis on its own until it determines that it has either turned back on itself, is about to try crossing another fringe or is outside a user-selected boundary, at which time it stops and asks for help from the operator. Since the trouble spots cannot be eliminated in advance or automatically compensated, this system requires virtually constant operator attention and is, therefore, not fully automatic, i.e., is only semi-automatic.
For many purposes, including inspection, robot vision, microscopy, computer-aided design (CAD), and computer aided manufacture (CAM), it would be ideal if there existed a fully automatic system which could unambiguously reconstruct the three-dimensional surface of an object from two-dimensional fringe patterns. In order to be fully automatic, such a system would have to be able to analyze touching or branching fringes.